We consider bearing estimation of multiple narrow-band plane waves impinging on an array of sensors. For this problem, bearing estimation algorithms such as minimum variance distortionless response (MVDR), multiple signal classification, and maximum likelihood generally require the array covariance matrix as sufficient statistics. Interestingly, the rank of the array covariance matrix is approximately equal to the number of the sources, which is typically much smaller than the number of sensors in many practical scenarios. In these scenarios, the covariance matrix is low-rank and can be estimated via matrix completion from only a small subset of its entries. We propose a distributed matrix completion framework to drastically reduce the inter-sensor communication in a network while still achieving near-optimal bearing estimation accuracy. Using recent results in noisy matrix completion, we provide sampling bounds and show how the additive noise at the sensor observations affects the reconstruction performance. We demonstrate via simulations that our approach sports desirable tradeoffs between communication costs and bearing estimation accuracy.